Understanding the concept of every open set being a disjoint union

[jdinatale]

of a countable collection of open intervals.

I'm having a hard time seeing how this could be true. For instance, take the open set (0, 10). I'm having a hard time seeing how one could make this into a union of countable open intervals.

For instance, (0,1) U (1, 10) or (0, 3) U (3, 6) U (6, 10) wouldn't work because those open intervals miss some points. There are "gaps" missing from the initial open set (0, 10). It seems like any union of DISJOINT intervals would have "gaps" missing from the initial open set. And if any of the open sets overlap to fill those gaps, then they are no longer disjoint.

I've read several proofs of this theorem, and they don't clear up my confusion.

 

[rubi]

An open set in $\mathbb R$ is by definition a (not necessarily disjoint) union of open intervals. You can show that $\mathbb R$ is second countable, so an at most countable number of open intervals suffices. Now a union of (not necessarily disjoint) open intervals is either an open interval again (for example $(0,2) \cup (1,3) = (0,3)$) or it is already a disjoint union of open intervals. So given an open set as a coutable union of not necessarily disjoint open intervals, you can always make it into a disjoint union of open intervals by joining the sets that have nonzero intersection.

In your case, $(0, 10)$ is already an open interval, so it is already a countable union of open intervals. Just take $A_0 = (0,10)$ and $A_n = \varnothing$ for $n\neq 0$.

 

[Erland]

Countable includes finite, so a finite set is also countable. So {(0,10)} is countable set and its union is (0,10), which is therefore a countable union of open intervals.

 

[A David]

of a countable collection of open intervals.

I'm having a hard time seeing how this could be true. For instance, take the open set (0, 10). I'm having a hard time seeing how one could make this into a union of countable open intervals.

For instance, (0,1) U (1, 10) or (0, 3) U (3, 6) U (6, 10) wouldn't work because those open intervals miss some points. There are "gaps" missing from the initial open set (0, 10). It seems like any union of DISJOINT intervals would have "gaps" missing from the initial open set. And if any of the open sets overlap to fill those gaps, then they are no longer disjoint.

I've read several proofs of this theorem, and they don't clear up my confusion.

You might be interested to know then that every open subset of the reals can be written as a countable union of almost disjoint closed intervals.

 

[blue_raver22]

Can you provide an explanation or clarification?

I understand your confusion and I can provide an explanation for this concept. The key to understanding this theorem lies in the definition of an open set. An open set is a set of points where any point within the set has a neighborhood that is also contained within the set. In other words, there are no "endpoints" or "boundary points" in an open set.

In the case of the open set (0, 10), it may seem like there are gaps because we are used to thinking about intervals in terms of their endpoints. But in reality, the open set (0, 10) does not have any endpoints. It is an unbounded set that includes all points between 0 and 10.

Now, let's consider the union of countable open intervals. This means that we are taking a collection of open intervals and combining them into one set. Each of these intervals will have its own "endpoints" or boundaries, but when we take the union, those boundaries will overlap and "fill in" the gaps in the open set (0, 10).

For example, let's take the open set (0, 10) and consider the following countable collection of open intervals:

(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10)

Each of these intervals has its own boundaries, but when we take the union, the boundaries overlap and create a continuous set that covers the entire open set (0, 10). This is the essence of the disjoint union theorem - we can take a countable collection of open intervals and combine them in such a way that they "fill in" the gaps in an open set.

I hope this explanation helps to clarify the concept of every open set being a disjoint union of a countable collection of open intervals. It may seem counterintuitive at first, but it is a fundamental concept in topology and has been proven to be true through rigorous mathematical proofs.